Inrecentyears,participatinglifeinsuranceproductsbecomemoreandmorepopularinmajorinsuranceandfinancemarketsaroundtheworld.Theseproductscanberegardedasinvestmentplanswithassociatedlifeinsurancebenefits,aspecifiedbenchmarkreturn,aguaranteeofanannualminimumrateof

(1)

Volume 2008, Article ID 474623,30pages doi:10.1155/2008/474623

Research Article

Pricing Participating Products under a Generalized Jump-Diffusion Model

Tak Kuen Siu,¹John W. Lau,²and Hailiang Yang³

1Department of Mathematics and Statistics, Curtin University of Technology, Perth, Western Australia 6845, Australia

2Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK

3Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong

Correspondence should be addressed to Tak Kuen Siu,t.siu@curtin.edu.au Received 21 January 2008; Accepted 20 May 2008

Recommended by Vo Anh

We propose a model for valuing participating life insurance products under a generalized jump- diﬀusion model with a Markov-switching compensator. It also nests a number of important and popular models in finance, including the classes of jump-diﬀusion models and Markovian regime- switching models. The Esscher transform is employed to determine an equivalent martingale measure. Simulation experiments are conducted to illustrate the practical implementation of the model and to highlight some features that can be obtained from our model.

Copyrightq2008 Tak Kuen Siu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In recent years, participating life insurance products become more and more popular in major insurance and finance markets around the world. These products can be regarded as investment plans with associated life insurance benefits, a specified benchmark return, a guarantee of an annual minimum rate of return, and a specified rule of the distribution of annual excess investment return above the guaranteed return. To enter the contract, policyholders pay annual premiums to an insurer, who will then manage and invest the funds in a specified reference portfolio. One key feature of these investment plans is the sharing of profits from an investment portfolio between the policyholders and the insurer. The specified rule of surplus distribution commonly used by insurers is known as reversionary bonus, which is employed to credit interest at or above a specified amount of guaranteed rate to the policyholders every year. The policyholders can receive an additional bonus at the maturity of

(2)

the contract, namely, the terminal bonus, if the terminal surplus of the fund is positive at the maturity. If the insurer defaults at the maturity of the policy, the policyholders can only receive the outstanding assets. For more comprehensive discussion on various features of participating policies, refer to Grosen and Jørgensen 1. Due to the internationally growing trend of adopting the market-based and fair valuation accountancy standards for the implementation of risk management practice for participating policies, it is practically important to develop appropriate, realistic, and objective models for valuing these policies.

Earlier works on exploring the use of the modern option pricing theory to value embedded options in with-profits life insurance policies go back to Brennan and Schwartz 2,3and Boyle and Schwartz4. Since then, there has been considerable interest on utilizing option pricing theory and its modern technologies to determine fair values of these policies.

Grosen and Jørgensen 1 develop a flexible contingent claims model to incorporate the minimum rate guarantee, bonus distribution, and surrender risk. Prieul et al. 5 adopt a partial differential equation approach to value a participating policy and employ the method of similarity transformations of variables to reduce the dimension of the partial differential equation governing the value of the policy. Bacinello 6, 7 adopt binomial schemes for computing the numerical solutions to the fair valuation problems of participating policies with various contractual features. Bacinello7introduces a model for describing the feature of annual premiums. Grosen and Jørgensen8use a barrier option framework to study and document the effect of regulatory intervention rules on reducing the insolvency risk of the policies. Chu and Kwok9 develop a flexible contingent claims model that describes rate guarantee, bonuses, and default risk. Siu10considers the pricing of a participating policy with surrender options when the market values of the reference portfolio are governed by a Markov-modulated geometric Brownian motion.

In this paper, we propose a model for valuing participating life insurance products under a generalized jump-diﬀusion model with a Markov-switching compensator. We suppose that the jump component is specified by the class of Markov-modulated kernel-biased completely random measures. The class of kernel-biased completely random measures is a wide class of jump-type processes. It has a very nice representation, which is a generalized kernel-based mixture of Poisson random measuresor, in general, random measures. The main idea of the kernel-biased completely random measure is to provide various forms of distortion of jump sizes of a completely random measure using the kernel function. This provides a great deal of flexibility in modeling diﬀerent types of finite and infinite jump activities compared with some existing models in the literature. We also provide additional flexibility to incorporate the impact of structural changes in macroeconomic conditions and business cycles on the valuation of participating policies by introducing an observable, continuous-time and finite- state Markov chain. Here the states of the Markov chain may be interpreted as proxies of some observable macroeconomic indicators, such as gross domestic product and retail price index.

They might also be considered economic ratings of a region or sovereign ratings. The model we considered here is general enough to nest a number of important and popular models for asset price dynamics in finance, including the two important classes of models, namely, the jump-diﬀusion models and the Markovian regime-switching models. These models are justified empirically in the literature and are shown to be practically useful for pricing and hedging derivatives. Our model can also be related to other important and popular classes of financial models, namely, the VG model pioneered by Madan et al.11and the CGMY model pioneered by Carr et al.12.

(3)

For valuing participating products under the generalized jump-diffusion model, we employ a well-known tool in actuarial science, namely, the Esscher transform, which provides a convenient and flexible way to determine an equivalent martingale measure under the incomplete market setting. We consider various special cases of the Markov-modulated kernel- biased completely random measure for the jump component, namely, the Markov-modulated generalized GammaMGGprocess, the scale-distorted version of the MGG process, and the power-distorted version of the MGG process. The MGG process encompasses the Markov- modulated weighted GammaMWGprocess and the Markov-modulated inverse Gaussian MIGprocess as special cases. We compare the fair values of the options embedded in the participating products implied by our generalized jump-diffusion models with those obtained from other existing models in the literature via simulation experiments and highlight some features of the qualitative behavior of the fair values that can be obtained from different parametric specifications of our model. The paper is outlined as follows.

Section 2 presents the generalized jump-diﬀusion model for the market value of the reference asset and the Esscher transform for valuation. We also provide some discussion for the hedging and risk management issues. InSection 3, we consider three important parametric cases of the Markov-modulated kernel-biased completely random measures, namely, the MGG, the scale-distorted and power-distorted versions of the MGG process. The simulation procedure and the simulation results of the fair values of the options embedded in the policy are presented and discussed inSection 4. The final section summarizes this paper. The proofs of the lemmas and propositions are presented in the appendix.

2. The valuation model

In this section, we consider a financial model consisting of a risk-free money market account and a reference risky asset or portfolio. We suppose that the market value of the reference asset is governed by a jump-diﬀusion model with the jump component being specified as a kernel-biased completely random measure with Markov-switching compensator. We assume that the market is frictionless and that the mortality risk and surrender option are absent.

We further impose certain assumptions on the rule of bonus distribution in our valuation model. We aim at developing a fair valuation model for participating life insurance policies which can incorporate the impact of the switching behavior of the states of the economy on the market value of the reference asset and fair value of the policy. The market described by the model is incomplete in generalsee13–16. Hence, there are infinitely many equivalent martingale measures and there is a range of no-arbitrage prices for a policy. Here, we determine an equivalent martingale measure by the Esscher transform. In the sequel, we introduce the set up of our model.

2.1. The price dynamics

In this subsection, we describe the price dynamics of the reference portfolio underlying the participating policy. Firstly, we fix a complete probability spaceΩ,F,P, wherePis the real- world probability measure. LetTdenote the time index set0, Tof the economy. We describe the states of the economy by a continuous-time Markov chain{Xt}_t∈TonΩ,F,Pwith a finite state spaceS: s1, s₂, . . . , s_N. Without loss of generality, we can identify the state space of the process{Xt}_t∈Tto be a finite set of unit vectors{e1, e2, . . . , eN}, whereei 0, . . . ,1, . . . ,0∈ R^N.

(4)

WriteQfor the generator orQ-matrixqiji, j1,2,...,N. Then, from Elliott et al.17, we have the following semimartingale decomposition for the process{Xt}t∈T:

XtX0

_t

0

QXsds Mt. 2.1

Here{Mt}_t∈Tis anR^N-valued martingale with respect to the filtration generated by{Xt}_t∈T. Let{rt, Xt}_t∈Tbe the instantaneous market interest rate of a bank account or a money market account, which depends on the state of the economy. That is,

r t, Xt

r, Xt

^N

i1

ri

Xt, ei

, t∈ T, 2.2

where r : r1, r2, . . . , rNwithri>0 for eachi1,2, . . . , Nand·,·denotes the inner product in the spaceR^N.

For notational simplicity, we writer_t forrt, Xt. In this case, the dynamics of the price process{Bt}_t∈Tfor the bank account is described by

dB_tr t, X_t

B_tdt,

B01. 2.3

In the sequel, we first describe a Markov-switching kernel-biased completely random measure.

James 18, 19 propose a kernel-biased representation of completely random measures, which provides a great deal of flexibility in modeling diﬀerent types of finite and infinite jump activities by choosing diﬀerent kernel functions. Here we employ the kernel-biased representation of completely random measures proposed by James 18,19 and adapt this representation to the Markov-modulated case in which the compensator of the underlying random measure switches over time according to the state of{Xt}_t∈T.

LetT,BTdenote a measurable space, whereBTis the Borelσ-field generated by the open subsets ofT. WriteB0for the family of Borel setsU ∈ R , whose closureUdoes not contain the point 0. LetXdenote T × R . The measurable space X,BXis then given by T × R ,BT⊗ B0.

For eachU ∈ B0, letN_X_t·, Udenote a Markov-switching Poisson random measure on the spaceX. WriteNXtdt, dzfor the diﬀerential form of the measureNXtt, U. LetρXtdz|t denote a Markov-switching L´evy measure on the spaceXdepending ontand the stateXt; ηis aσ-finitenonatomicmeasure onT. Note that ifX_t e_ii1,2, . . . , N, writeρ_i:ρ_e_idz|t.

To ensure the existence of the kernel-biased completely random measure to be defined in the sequelsee18–20, we assume that for an arbitrary strictly positive function on R , h, ρi, andηare selected in such a way that for each bounded setBinT,

N i1

B

R min hz,1

ρidz|tηdt<∞. 2.4

We assume that the Markov-switching intensity measureνXtdt, dzfor the Poisson random measureN_X_tdt, dzis given by

νXtdt, dz:ρXtdz|tηdt ^N

i1

ρidz|t Xt, ei

ηdt. 2.5

(5)

By modifying the kernel-biased representation of James 18, 19, we define a Markov- modulated kernel-biased completely random measureμXtdtonTas follows:

μXtdt:

RhzNXtdt, dz, 2.6 which is a kernel-based mixture of the Markov-modulated Poisson random measure N_X_tdt, dzover the state space of the jump sizeR with the mixing kernel functionhz. See also Perman et al.20for discussion on representations of completely random measures. In general, we can replace the Poisson random measure with a random measure and choose some quite exotic functions forhzto generate diﬀerent types of finite and infinite jump activities.

Letm_X_t denote the mean measure ofμ_X_t. That is,

mXtdt

R hzνXtdt, dz ^N

i1

R hzρidz|t Xt, ei

ηdt . 2.7

Let {Wt}_t∈T denote a standard Brownian motion onΩ,F,Pwith respect to the P- augmentation of its natural filtrationF^W :{F^W_t }_t∈T. We suppose thatW, X, andμ_X_tdtare independent. LetNXtdt, dzdenote the compensated Poisson random measure defined by

N_X_tdt, dz:N_X_tdt, dz−ρ_X_tdz|tηdt. 2.8

Let μ_t and σ_t denote the drift and volatility of the market value of the reference asset, respectively. We suppose thatμtandσtare given by

μt: μ, Xt

^N

i1

μi

Xt, ei

,

σt: σ, Xt

^N

i1

σi Xt, ei

,

2.9

whereμ: μ1, μ2, . . . , μNandσ: σ1, σ2, . . . , σN; μi∈ Randσi>0, for eachi1,2, . . . , N.

Then, we assume that the dynamic of the market valueAof the reference portfolio is governed by the following general geometric jump-diﬀusion process with a Markov-switching kernel-biased completely random measure:

dAt At−

μtdt−mXtdt σtdWt

Re^hz−1NXtdt, dz

. 2.10

By convention, we suppose thatA01,P-a.s. Similar to the Merton jump-diﬀusion model, the drift term ofAis given by the meanμ_tdtminus the Markov-switching compensatorm_X_tdtof μ_X_tdt. We can then write the dynamic ofAas follows:

dA_tA_t−

μ_tdt

R

e^hz−1−hz

ρ_X_tdz|tηdt σ_tdW_t

R

e^hz−1N_X_tdt, dz

. 2.11

(6)

In general, one can consider the situation that the driftμtand the volatilityσtdepend not only on the current economic stateXt, but also other state variables or market information, such as the current value of the reference portfolioA_t, when dealing with a long-term maturity. This represents an interesting and practically relevant direction for further generalizing the model.

To focus on modeling and examining the impact of transitions of economic states on the price dynamics of the reference portfolio and the fair value of the policy, we assume here thatμtand σ_tdepend on the current economic stateX_tonly.

LetYt :lnAt. Note thatY00,P-a.s., sinceA01. Then, by It ˆo’s formula,

dY_t

μ_t−1

2σ_t² dt σ_tdW_t

RhzN_X_tdt, dz. 2.12 2.2. The crediting scheme

Now, we describe the scheme for evaluating the interest rate credited to the policy reserve. Let Rt denote the book value of the policy reserve andDtthe bonus reserve, at timet∈ T. Then, as in Chu and Kwok9, we have the following accounting identity forA_t,R_t,andD_t:

A_t R_t D_t, t∈ T, 2.13

whereR₀ :α_pA₀, α_p∈0,1, andR₀is interpreted as the single initial premium paid by the policyholder for acquiring the contract andαpis the cost allocation parameter. In this case, the αp-portion of the initial asset portfolio is financed by the policyholder.

Writec_RA, Rfor the interest rate credited to the policy reserve. Then we have

dR_t c_RA, RRtdt. 2.14 In practice, the specification ofc_RA, Rdepends on the rule of bonus distribution, which is decided by the management level of an insurance company. Typically, an insurer distributes to his/her policyholder a certain proportion, sayδ, of the excess of the ratio of bonus reserve Dt to the policy reserveRt over the target ratioβ, which is a long-term constant target ratio specified by the management. The proportional constant δ is called the reversionary bonus distribution rate and it is assumed thatδ∈0,1. For the crediting scheme of interest rate, it is also assumed that there is a specified guarantee ratergfor the minimum interest rate credited to the policyholder’s account. This means that the interest ratec_RA, R≥r_g. Here, we adopt the interest rate crediting scheme used in Chu and Kwok9as follows:

c_R A_t, R_t

max

r_g,

lnA_t R_t −β

, 2.15

where the interest ratecRAt, Rtcredited to the policyholder’s account depends on both the reversionary bonusβand the guaranteed raterg.

2.3. Pricing by the Esscher transform

In this subsection, we describe how to determine an equivalent martingale measure by the Esscher transform in the incomplete market specified by the generalized jump-diﬀusion

(7)

model. Firstly, we provide a short discussion on other existing approaches for option valuation in an incomplete market.

Different approaches have been proposed in the literature on how to pick an equivalent martingale pricing measure in an incomplete market. F öllmer and Sondermann21, F öllmer and Schweizer 22 introduce the notion of a minimal martingale measure and select a unique equivalent martingale measure via risk-minimization. Duffie and Richardson23and Schweizer24propose the mean-variance criterion for determining an equivalent martingale measure. Davis25 adopts the marginal rate of substitution, which is a sound equilibrium argument in economic theory, to pick a pricing measure by solving a utility maximization problem. The pioneering work by Gerber and Shiu26provides a pertinent solution to the option pricing problem in an incomplete market by the Esscher transform, a time-honored tool in actuarial science introduced by Esscher27. The Esscher transform provides market practitioners with a convenient and flexible way to value options. Here, we employ the regime- switching Esscher transform in the work of Elliott et al. 16 and present the idea of this transform in the sequel.

Firstly, we describe the information structure of the model. LetF^X:{F^X_t }_t∈TandF^Y : {F^Y_t}_t∈Tdenote theP-augmentation of the natural filtration generated byXandY, respectively.

For eachi1,2 andt∈ T, writeGtfor theσ-algebraF^X_T∨ F^Y_t . LetBMTdenote the collection ofBT-measurable and nonnegative functions with compact support onT. WriteBTfor the Borelσ-field ofT. For each processθ∈BMT, writeθ·Y_tfor_t

0θ_udY_u, for eacht∈ T. Let MYθ_t:EPe^−θ^·^Y^t | F^X_T, whereEPrepresents expectation underP.

Let{Λt}_t∈Tdenote aG-adapted stochastic process defined as below:

Λt: e^−θ^·Y^t

MYθ_t, t∈ T. 2.16

Applying It ô’s differentiation rule for jump-diffusion processessee, e.g.,28,29, we have

e^−θ^·Y^t 1− _t

0

e^−θ^·Y^sθs

μs−1

2σ_s² ds− _t

0

e^−θ^·Y^sθsσsdWs

− _t

0

Re^−θ^·^Y^s−θs−hzNXsds, dz 1 2

_t

0

e^−θ^·^Y^sθ²_sσ_s²ds _t

0

Re^−θ^·^Y^s−

e^−θ^s^hz−1NXsds, dz _t

0

Re^−θ^·^Y^s−

e^−θ^s^hz−1 θ_shz

ρ_X_sdz|sηds.

2.17

Conditioning onF^X_T for both sides of2.17,

E

e^−θ^·Y^tF^X_T 1−

_t

0

E

e^−θ^·Y^sF^X_T θs

μs−1

2σ_s² ds 1 2

_t

0

E

e^−θ^·^Y^sF_T^X θ²_sσ_s²ds _t

0

R E

e^−θ^·^Y^s−F^X_T

e^−θ^s^hz−1 θ_shz

ρ_X_sdz|sηds.

2.18

(8)

Hence, MYθ_t exp

− _t

0

θs

μs−1

2σ_s² ds 1 2

_t

0

θ_s²σ_s²ds _t

0

R

e^−θ^s^hz−1 θshz

ρXsdz|sηds

. 2.19 Therefore,

Λtexp

− _t

0

θ_sσ_sdW_s−1 2

_t

0

θ²_sσ_s²ds− _t

0

Rθ_shzN_X_sds, dz

− _t

0

R

e^−θ^s^hz−1 θshz

ρXsdz|sηds .

2.20

Lemma 2.1. Λis aG,P-martingale.

Then, the Esscher transformQ∼PonGtwith respect to{θt |t∈ T}is defined as dQ

dP _G

t

Λt, t∈ T. 2.21

Harrison and Kreps30and Harrison and Pliska31,32establish the relationship between the absence of arbitrage opportunities and the existence of an equivalent martingale measure.

This is called the fundamental theorem of asset pricing. Delbaen and Schachermayer33point out that the equivalent relationship does not hold “true” in general and show that the absence of arbitrage is “essentially” equivalent to the existence of an equivalent martingale measure under which the discounted stock price process is a martingale. WriteA_t :exp−_t

0r_uduAt. In our setting, the martingale condition is given by

AsEQ At|Gs

, for anyt, s∈ Twitht≥s, 2.22

whereE_Qrepresents expectation underQ.

Proposition 2.2. Suppose there exists a functionη·:0, T→R such thatηdt ηtdt. Then, the martingale condition is satisfied if and only ifθ_tsatisfies

μt−rt−θtσ_t²

R

e^−θ^t^hz

e^hz−1

−hz

ρXtdz|tηt 0, 2.23 for eacht∈ T.

Proposition 2.3. LetW :{Wt∈T}denote a standard Brownian motion and letN^Q_X

tdt, dzdenote a compensated Markov-modulated Poisson random measure with compensatorρ^Q_X

tdz|tηtdtunderQ, whereρ_X^Q

tdz|t:e^−θ^t^hzρXtdz|t. Then, underQ, dY_t

r_t−1

2σ_t² dt

R

1−e^hz hz ρ_X^Q

tdz|tηtdt σ_tdW_t

R hzN_X^Q

tdt, dz, 2.24 whereXis governed by2.1.

(9)

2.4. Fair valuation

Here, we present the procedure for the fair valuation based on an equivalent martingale measure chosen by the regime-switching Esscher transform in the last subsection.

LetVAT, R_T, X_Tdenote the terminal payoﬀof the participating policy on the policy’s maturity dateT, when the state of the economyXTat timeTisX. Then,

V

AT, RT, XT

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

A_T, ifA_T < R_T, RT, ifRT≤AT≤ RT

αp, RT γP1T, ifAT >RT

αp,

2.25

whereγ is the terminal bonus distribution rate andP_1T : maxαpA_T−R_T,0is the terminal bonus option.

Let P2T : maxRT −AT,0, whereP2T represents the terminal default option on the policy’s maturity date T. Then, the terminal payoﬀ VAT, RT, XT can be written in the following form:

V

A_T, R_T, X_T

R_T γP_1T−P_2T. 2.26 Note that the bonus option can be viewed as a standard European call option that grants the policyholder the right to pay the policy value as a strike price to receive αp-portion of the asset portfolio. Instead of evaluating the fair value of the terminal payoﬀof the policy, we consider the fair valuation for each of the components of the terminal payoﬀof the policy, namely, the guaranteed benefit R_T, the terminal bonus option P_1T,and the terminal default optionP2T. Given knowledge ofGt, the conditional fair values of the guaranteed benefit, the terminal bonus option, and the terminal default option at timetare, respectively,

Gt EQ

exp

− _T

t

rsds RT

Gt

,

P₁t E_Q

exp

− _T

t

r_sds P_1T Gt

,

P2t EQ

exp

− _T

t

rsds P2T

Gt

.

2.27

Note that the discount factor here is stochastic and switches over time according to the states of the Markov chain.

As in Buﬃngton and Elliott14,15, given thatA_t A, R_t R andX_t X, the fair values of the guaranteed benefit, the terminal bonus option, and the terminal default option at timetare, respectively,

G

A_t, R_t, X_t E_Q

exp

− _T

t

r_sds R_T|A_tA, R_tR, X_tX

, 2.28

P1

At, Rt, Xt

EQ

exp

− _T

t

rsds P1T|AtA, RtR, XtX

, 2.29

P₂

A_t, R_t, X_t EQ

exp

− _T

t

r_sds P_2T|A_tA, R_t R, X_t X

. 2.30

(10)

2.5. Hedging and risk management

Besides fair valuation of the options embedded in the participating policy, it is interesting to investigate how the risks inherent in these options can be hedged once the policy has been sold from a risk management perspective. The main focus of the current paper is the fair valuation issue of the policy. In practice, the hedging and risk management issues of the policy are also important. So, we provide some discussion for the hedging and risk management issues of the policy here. The hedging and risk management issues of the policy are certainly interesting and important topics for future research.

There are different ways to hedge the risks inherent in the options embedded in the policy. Hedging via the Greeks and the risk-minimizing hedging represent two popular approaches to hedging these risks. However, due to the fact that the market model considered here is incomplete, perfect hedging cannot be achieved. Here we discuss the use of the Greeks to hedge the risks inherent in the options, namely, the guaranteed benefit, the terminal bonus option, and the default option, embedded in the policy. Note that hedging using the Greeks is only an approximating hedging strategy and that it cannot provide a perfect hedging result due to the market incompleteness. There are different approaches to compute the Greeks based on the Monte Carlo simulation of the price paths. The basic method is the Monte Carlo finite-difference approach. The key idea of this method is to compute the finite difference approximation of the differentials using the Monte Carlo simulation. For illustration, we consider the use of this method to compute the Delta. Suppose VAandVA denote Monte Carlo estimators of “true” pricesVAandVA , respectively, whereArepresents the initial value of the reference portfolio andis asmallpositive constant. Then, the Delta ΔAof an option evaluated at the initial value Acan be estimated by the finite-difference estimator as follows:

ΔA VA −VA

. 2.31

Glynn34shows that if the simulations of the two estimatorsVAandVA are drawn independently, the best possible convergence rate isn^−1/4, wherenis the number of simulation runs. The convergence rate can be improved using the central diﬀerenceVA −VA− /2. In this case, the best possible convergence rate isn^−1/3. The convergence rate can further be improved using common random numbers for both Monte Carlo estimators. The optimal convergence rate one can achieve in this case isn^−1/2, which is the same as the best possible convergence rate of a crude Monte Carlo method.

Other approaches that enhance the efficiency of the computation of the Greeks based on the Monte Carlo simulation include the simple differentiation approach proposed by Broadie and Glasserman35and the Malliavin calculus approach discussed by Fournié et al.36,37.

Chen and Glasserman38nvestigate the connection between the Malliavin calculus approach and the traditional approach based on the pathwise method and likelihood method. Recently, the Malliavin calculus approach for the Monte Carlo computation of the Greeks for jump- diffusion models and Lévy processes has been developed by several authors, including Le ón et al.39, El-Khatib and Privault40, Davis and Johansson41, and others. It is interesting to explore how the Malliavin calculus approach for jump-diffusion processes and Lévy processes can be extended to deal with the hedging of the risks inherent in the options embedded in the participating policy under the Markovian regime-switching jump-diffusion model considered

(11)

here. Besides using the Malliavin calculus approach, one may also consider the use of an extended Clark-Haussman-Ocone formula using the white noise analysis in the work of Aase et al. 42 to hedge the risks of the options embedded in the policy under the Markovian regime-switching jump-diﬀusion model. This also represents an interesting topic for further research.

3. Various parametric specifications to the jump component

In the previous section, we have defined a general jump-diﬀusion process with the jump component specified by a kernel-biased Markov-modulated completely random measure.

Here, we consider some parametric cases of the general jump process by specifying some particular forms of the kernel function and the Markovian regime-switching intensity measure.

These parametric cases include the MGG process, the scale-distorted and power-distorted versions of the MGG process, and their special cases. We also derive the risk-neutral dynamics for the logarithmic return process{Yt}_t∈TunderQfor various parametric specifications which will be used for computing the fair values of the policies inSection 4. It is interesting to note that the kernel-biased completely random measure has some connections to some important L´evy processes in the literature including the VG process by Madan et al.11and the CGMY model of Carr et al.12. We also discuss these connections in this section.

3.1. Markov-modulated generalized Gamma (MGG) process

The generalized GammaGGprocess is a wide class of jump-type processes, which consists of the weighted GammaWGprocess and the inverse GaussianIGprocess as special cases.

The GG process is a special case of the kernel-biased completely random measure and can be obtained by setting the kernel functionhz zand choosing a particular parametric form of the compensator measure. To provide more flexibility in describing the impact of the states of an economy on the jump component, we consider a Markov-modulated GG process, called the MGG process, whose compensator switches over time according to the states of the economy.

We first describe the MGG process in the sequel.

Letα≥0 denote a constant shape parameter of the MGG process. We suppose that the scale parameter of the MGG processbt:bt, Xtswitches over time according to the states of the Markov chainXand is given by

bt: b, Xt

^N

i1

bi Xt, ei

, 3.1

where b : b1, b₂, . . . , b_N∈ R^Nandb_i≥0, for eachi1,2, . . . , N.

Then, the Markov-switching intensity process of the MGG process is ρ_X_tdz|tηtdt 1

Γ1−αe^−b,X^t^zz^−α−1dzηtdt^N

i1

1

Γ1−αe^−bⁱ^zz^−α−1 X_t, e_i

dzηtdt.

3.2 In this case, the martingale condition becomes

μ_t−r_t−θ_tσ_t²

R

e^−θ^t^z e^z−1

−z

ρ_X_tdz|tηt 0, 3.3 whereρXtdz|tηtis given by3.2.

(12)

Writeρ^Q_X

tdz|t:e^−θ^t^zρXtdz|t, whereθtsatisfies3.3. LetN^Q_X

tdt, dzdenote a Poisson random measure with Markov-switching compensatorρ^Q_X

tdz|tηtdtunderQ. Then, under Q, the dynamic ofYis

dY_t r_t−1

2σ_t² dt

R

1−e^hz ρ_X^Q

tdz|tηtdt σ_tdW_t

R hzN_X^Q_tdt, dz. 3.4 Whenα0, the MGG process reduces to a Markov-modulated WGMWGprocess. That is, the Markov-switching intensity of the MWG process is

ρXtdz|tηtdt e^−b,X^t^z

z dzηtdt^N

i1

e^−bⁱ^z z

Xt, ei

dzηtdt. 3.5

In this case, the martingale condition becomes

R

e^−θ^t^z e^z−1

−ze^−b,X^t^z

z dzηt 0. 3.6

UnderQ, dY_t

r_t−1

2σ_t² dt

R

1−e^hze^−θ^t^b,X^t^z

z dzηtdt σ_tdW_t

RhzN_X^Q_tdt, dz, 3.7 whereN_X^Q

tdt, dzis a Poisson random measure with Markov-switching compensator, ρ^Q_X

tdz|tηtdte^−θ^t^b,X^t^z

z dzηtdt, 3.8

andθtsatisfies3.6.

Whenα1/2, the MGG becomes a Markov-modulated IGMIGprocess. In this case, the martingale condition becomes

R

e^−θ^t^z e^z−1

−z 1 Γ1/2√

ze^−b,X^t^zdz dzηt 0. 3.9 UnderQ, the dynamic ofYis

dYt rt−1

2σ_t² dt

R

1−e^hz 1 Γ1/2√

ze^−θ^t^b,X^t^z dzηtdt σtdWt

R hzN_X^Q_tdt, dz,

3.10

where the intensity process forN_X^Q

tdt, dzis ρ^Q_X

tdz|tηtdt 1 Γ1/2√

ze^−θ^t^b,X^t^z dzηtdt, 3.11 andθtsatisfies3.9.

(13)

3.2. The scale and power distortions of the MGG process

In this section, we consider the scale-distorted and power-distorted versions of the MGG process. The scale-distorted and power-distorted versions of the MGG process provide additional flexibility for describing various jump-type behaviors. They can describe the overstate and understate of jump amplitudes due to overreaction and underreaction of market participants to extraordinary events, respectively. For the scale-distorted version of the MGG process, the kernel functionhz cz, where c is a positive constant. Whenc >1, jump sizes are overstated. When 0< c < 1, jump sizes are understated. For the power-distorted version of the MGG process, the kernel functionhz z^q, whereq > 0. When q > 1, small jump sizesi.e., 0 < z < 1are understated and large jump sizesi.e.,z >1are overstated. When 0 < q < 1, small jump sizes are overstated and large jump sizes are understated. The scale- distorted and power-distorted versions of the MGG process with diﬀerent scale and power- distorted parameters can generate diﬀerent types of behaviors of market participants when they react to extraordinary events. They can shed lights on understanding the impact of these market participants’ behaviors on the price dynamic of the reference asset from a behavioral finance perspective. In general, one can also assume that the scalec for the scale-distorted version of the MGG process and the powerq for the power-distorted version of the MGG process also switch over time according to the state of the Markov chain. In this case, the overstate and understate of the jump amplitudes also depend on the state of the economy.

Here, we consider the case that bothcandqare constants for illustration.

For both the scale-distorted and power-distorted versions of the MGG process, the Markov-switching intensity processes are the same as that of the MGG process. For the scale- distorted version of the MGG process, the martingale condition is given by

R

e^−θ^t^cz e^cz−1

−cz

ρXtdz|tηt 0, 3.12

whereρXtdz|tηtis given by3.2.

UnderQ, the dynamic ofYis

dY_t

r_t−1 2σ_t² dt

R

1−e^cz ρ^Q_X

tdz|tηtdt σ_tdW_t

RczN_X^Q

tdt, dz, 3.13

whereρ^Q_X

tdz|t e^−θ^t^zρXtdz|tηdtandρXtdz|tηtdtis given by3.2.

For the power-distorted version of the MGG process, the martingale condition is

R

e^−θ^t^z^q e^z^q−1

−z^q

ρ_X_tdz|tηt 0, 3.14

whereρ_X_tdz|tηtis specified by3.2.

UnderQ, the dynamic ofYis

dYt

rt−1 2σ_t² dt

R

1−e^z^q ρ^Q_X

tdz|tηtdt σtdWt

R z^qN_X^Q

tdt, dz, 3.15

whereρ^Q_X

tdz|t e^−θ^t^zρXtdz|tandρXtdz|tηtis given by3.2.

(14)

Whenα0, the scale-distorted version of the MGG process becomes the scale-distorted version of the MWG process and the power-distorted version of the MGG process reduces to the power-distorted version of the MWG process. Whenα 1/2, the scale-distorted and power-distorted versions of the MGG process become the scale-distorted and power-distorted versions of the MIG process, respectively.

3.3. Connections to the VG and CGMY processes

Now, we outline some connections of a modified version of the Markov-modulated kernel- biased completely random measure to the VG and CGMY processes. We first provide some discussions on the VG model. The VG process can be represented in a number of equivalent ways, namely, the representation based on the time-changed Brownian motion, the difference between two gamma processes, the Lévy measure representation, the predictable compensator representation, where the predictable compensator representation is closely related to the Lévy measure representation in a fundamental way. Madan et al.11and Elliott and Royal43 provide detailed discussion for different representations of the VG process. It has been shown by Elliott and Royal43that the predictable compensator of the VG process is the same as the Lévy measure. The Lévy measure of the VG process is given bysee11

ν^VGdz, dt k^VGzdz dt

Cexp−Mz

z I{z>0}−CexpGz

z I{z<0} dz dt, 3.16 whereC, M, G∈ R are parameters of the VG process.

We consider a Markov-modulated version of the VG process with the following Markov- switching compensator:

ν^VG_X

t dz, dt k_X^VG

t zdz dt

Cexp

− M, Xt

z

z I{z>0}−Cexp G, Xt

z

z I{z<0} dz dt,

3.17

where M : M1, M₂, . . . , M_N∈ R^Nand G : G1, G₂, . . . , G_N∈ R^NwithM_i >0 andG_i>0, for eachi1,2, . . . , N.

The Markov-modulated VG process can be related to a modified version of the Markov- switching kernel-biased completely random measure by suitable matching of the model parameters. Consider two Markov-switching Poisson random measuresN_X^k

tdz, dt, k 1,2, with the following intensity processes:

ρ^k_X

tdz|tηtdt e⁻¹^k^b^k^,X^t^z

z dzηtdt, 3.18

where b^k: b^k₁, b^k₂, . . . , b^k_N∈ Rwithb^k_i >0, for eachi1,2, . . . , N.

WriteNXtdz, dt:N_X¹

tdz, dtI{z>0} N_X²

tdz, dtI{z<0}. Then, the intensity process of N_X_tdz, dtis

ρXtdz|tηtdt

e^−b¹^,X^t^z

z I{z>0}−e^b²^,X^t^z

z I{z<0} dzηtdt. 3.19

(15)

Suppose, for eachk1,2, thathk·:R →R satisfies the following condition:

N i1

B

R min

h_kz,1

ρ^k_idz|tηdt<∞. 3.20

Leth·:R→Rbe a real-valued function defined as follows:

hz h₁zI{z>0}−h₂−zI{z<0}. 3.21 Then, define a processμtas follows:

μt: _t

0

RhzNXudz, du. 3.22 Letηdt dti.e.,η·is a uniform density. In this case,ηt 1. We further assume that b_i¹ M_i; b²_i G_i; C 1. Then, the Markov-modulated VG process with Markov-switching compensatorν^VG_X

t dz, dtcoincides with the processμt, which is a modified version of the Markov-switching kernel-biased completely random measure.

In the sequel, we consider the CGMY process. From Carr et al.12, the L´evy measure of the CGMY process is

ν^CGMYdz, dt k^CGMYzdz dt

Cexp

−G|z|

|z|^{1 Y} I{z<0} Cexp

−M|z|

|z|^{1 Y} I{z>0} dz dt,

3.23

whereC >0, G≥0, M≥0,andY <2.

We consider a Markov-modulated version of the CGMY process with the following Markov-switching compensator:

ν_X^CGMY_t dz, dt k^CGMY_X_t zdz dt

Cexp

− G, X_t

|z|

|z|^{1 Y} I_{z<0} Cexp

− M, X_t

|z|

|z|^{1 Y} I_{z>0} dz dt,

3.24

where M : M1, M2, . . . , MN∈ R^Nand G : G1, G2, . . . , GN∈ R^NwithMi >0 andGi>0, for eachi1,2, . . . , N.

For eachk 1,2, letN^k_X_tdz, dtdenote a Markov-switching Poisson random measure with the following intensity process:

ρ_X^k

tdz|tηtdt 1

Γ1−αe^−b^k^,X^t^zz^−α−1dzηtdt 1

Γ1−α

e^−b^k^,X^t^|z|

|z|^{1 α} dzηtdt.

3.25

LetNXtdz, dt:N¹_X_tdz, dtI{z>0} N²_X_tdz, dtI{z<0}. Then, the Markov-switching intensity process forNXtdz, dtis

ρ_X_tdz|tηtdt 1

Γ1−α

e^−b¹^,X^t^|z|

|z|^{1 α} I{z>0} 1 Γ1−α

e^−b²^,X^t^|z|

|z|^{1 α} I{z<0} dzηtdt. 3.26